
J. DOUGLAS HAMILTON DICKSON ON LEAST ROOTS OF EQUATIONS. 133 
Eliminating ¢,,....¢, from m of these equations and equation (10) we have 
Pam E Loree 3 wnt HE Ai | == ie Gt) 
EX a ee > Oe CANO Aigeeenan a aa 
aN est os ee ee Aye AS eal 
Aaa tg tan At ee 
an equation of the m” degree which gives the least root. 
But this equation possesses m — 1 roots in addition, and it will now be shown 
that these are the next m— 1 roots in order of ascending magnitude. 
Let this determinant equation be written thus 




g(a) g(a) Le 
F(@),a” > F(a)? +1 a ao 
p (7) p (s) G (7) p ( s) 
F(r)re 1 F(s)s? tnt sah E'(r)rP +1 +. FG@).2 Fl? °° 
Subtract the first horizontal line from the second, z times the first from the 
third, 2’ times the first from the fourth, and so on; and a/ter having done so, 
write 7 for z. The determinant will now be of the form 
pee), (gr) as 
Er)? > P(r)? eal 
9(s) g(t) g(s) g(t) 


Fis tPO@® +> Fort? Faeit ++ 
i.¢., When expanded it will contain 7 on/y where x was in the previous equation; 
and as the form remains the same, this equation will vanish if s be written for 
r when p is infinite. But s is the next greater root of F(z)=0. Thus the two 
least roots of F(v)=0 are roots of the equation (11). In exactly the same 
manner, ¢ and the succeeding higher roots of F(x)=0 would be roots of 
equation (11), until m of them had been employed; or in other words, the m 
least roots of F(z) =0 are given by the determinant equation (11) of the m+1™ 
order. 
It is scarcely necessary to add that the process applied to the determinant 
quadratic when the two least roots were equal will also apply here. In fact 
_ the equation holds independently of the equality of the roots. 
| Thus, if the coefficients found by ordinary division in equation (2) be 
_ employed to construct equation (11), the solution of this equation will give the 
| m least roots of F(z) =0. 
